ON TRANSPORT IN POROUS FORMATIONS CHARACTERIZED BY HETEROGENEITY OF EVOLVING SCALES

Solute transport in natural formations at the regional scale is influe nced by several scales of heterogeneity which correspond to the presen ce of several geological units called facies. As customarily assumed i n stochastic theories, inside the facies the transport can be characte rized by a single scale of heterogeneity. At the regional scale Severa l geological units are present such that a hierarchy of relevant scale s needs to be defined. A possible model for this spatial variability a ssumes the log conductivity as a random space function of stationary i ncrements characterized by a power law semivariogram. With this hypoth esis the ergodic dispersion coefficient grows unbounded as time increa ses, leading to the phenomenon called anomalous dispersion, An alterna tive approach considers the plume in nonergodic conditions and assumes the effective dispersion coefficient, which is defined through differ entiation in time of the expected value of the spatial second-order pl ume moment, as representative of macrodispersion. Large differences ha ve been observed in the resulting plume spreading while approaching th e problem using the above alternative definitions. In this paper we pr ovide first-order analytical solutions for the longitudinal effective dispersion coefficient, D-L, as well as for the expected value of the longitudinal spatial plume moment, [S-11] that complement semianalytic al expressions recently proposed in literature. Furthermore, we provid e a semianalytical expression for the standard deviation of the longit udinal second-order moment which is important in assessing the interva l of confidence of the estimation provided by [S-11]. Suitable numeric al simulations are performed to validate analytical and semianalytical expressions as well as to assess the impact of the cutoff in the log conductivity power spectrum imposed by choosing a finite domain dimens ion, We conclude that according to recently published results, the dis persion is anomalous when the Hurst coefficients, H, is larger than 0. 5 while it is Fickian for H < 0.5. This is in contrast with the ergodi c analysis which concludes that the dispersion is anomalous irrespecti ve of the Hurst coefficient, Hence the effective dispersion coefficien t is more effective than the ergodic dispersion. coefficient to repres ent the plume spreading. However, the standard deviation of the longit udinal spatial second-order moment is of the same order of magnitude a s the expected value leading to the conclusion that the estimations pr ovided by D-L and [S-11] are affected by large uncertainties. Numerica l results are in good agreement with the analytical solutions, and und er some hypotheses they are not influenced by the cutoff. This is not the case for the ergodic second-order longitudinal moment, which stron gly depends on the imposed cutoff.